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Discrete distributions<br />
(sample space is a subset of the integers)<br />
Discrete uniform(a, b) a, b ∈ Z, a < b<br />
Equally likely events labelled with integers<br />
from a to b.<br />
PMF<br />
Support<br />
Mean<br />
Variance<br />
1/n n = b a +1<br />
{a, a +1,...,b 1,b}<br />
a + b<br />
2<br />
n 2 1<br />
12<br />
Bernoulli(p) p ∈ [0, 1]<br />
An event with two outcomes:<br />
1 = success, 0 = failure<br />
PMF<br />
Support<br />
MGF<br />
Mean<br />
Variance<br />
1 p x =0<br />
{0, 1}<br />
p x =1<br />
1 p + pe t<br />
p<br />
p(1 p)<br />
Binomial(n, p) n = 1, 2, 3, ...; p ∈ [0, 1]<br />
The number of successes in n Bernoulli(p)<br />
trials (a count)<br />
PMF<br />
✓ ◆<br />
n<br />
p<br />
x<br />
x (1 p)<br />
Support<br />
MGF<br />
Mean<br />
Variance<br />
{0, 1,...,n}<br />
(1 p + pe t ) n<br />
np<br />
np(1 p)<br />
n x<br />
Geometric(p) p ∈ [0, 1]<br />
The number of successes in Bernoulli(p)<br />
trials until one failure.<br />
PMF<br />
Support<br />
MGF<br />
Mean<br />
Variance<br />
(1 p)p x<br />
{0, 1, 2,...}<br />
1 p<br />
1 pe t<br />
p<br />
1 p<br />
p<br />
(1 p) 2<br />
Negative binomial(r, p) r = 1, 2, 3, ... p ∈ [0, 1]<br />
The number of successes in Bernoulli(p)<br />
trials until r failures.<br />
PMF<br />
✓<br />
x + r<br />
x<br />
1<br />
Support<br />
MGF<br />
Mean<br />
Variance<br />
{0, 1, 2,...}<br />
✓<br />
1 p<br />
1 pet ◆r p<br />
r<br />
1 p<br />
p<br />
r<br />
(1 p) 2<br />
◆<br />
(1 p) r p x<br />
Poisson(λ) λ > 0<br />
Count of events that have constant rate,<br />
and are independent of one another.<br />
PMF<br />
Support<br />
MGF<br />
Mean<br />
Variance<br />
x e<br />
x!<br />
{0, 1, 2,...}<br />
exp( (e t<br />
1))
Continuous distributions<br />
(sample space is an interval on the real line)<br />
Uniform(a, b) a, b ∈ R<br />
Likelihood of event proportional to it’s<br />
length.<br />
PDF<br />
CDF<br />
Support<br />
Mean<br />
Variance<br />
1<br />
b a<br />
x a<br />
b a<br />
[a, b]<br />
a + b<br />
2<br />
(b a) 2<br />
12<br />
Exponential(θ) θ > 0<br />
Waiting time until a Poisson event with<br />
average waiting time θ.<br />
PDF<br />
CDF<br />
Support<br />
MGF<br />
Mean<br />
Variance<br />
1 e x/<br />
1 e x/<br />
[0, 1)<br />
1<br />
1 t<br />
✓<br />
✓ 2<br />
Exponential(λ) λ > 0<br />
Waiting time until a Poisson event with rate<br />
λ.<br />
PDF<br />
Mean<br />
1/<br />
e x<br />
Gamma(α, θ) α > 0, θ > 0<br />
Waiting time for α Poisson events with<br />
average waiting time θ.<br />
PDF<br />
CDF<br />
Support<br />
MGF<br />
Mean<br />
Variance<br />
⇥<br />
( ) x<br />
1 e x/⇥<br />
No closed form<br />
[0, 1)<br />
1<br />
(1 t) ↵<br />
↵✓<br />
↵✓ 2<br />
Gamma(α, β) β > 0, θ > 0<br />
Waiting time for α Poisson events with rate<br />
β.<br />
PDF<br />
Mean<br />
⇥<br />
( ) x<br />
/⇥<br />
1 e ⇥x<br />
Normal(μ,σ 2 ) μ ∈ R, σ 2 > 0<br />
PDF<br />
CDF<br />
Support<br />
MGF<br />
Mean<br />
Variance<br />
1<br />
p 2 ⇥ e<br />
(x)<br />
( 1, 1)<br />
exp(µt + 1<br />
2<br />
µ<br />
2<br />
(x µ)2<br />
2 2<br />
2 t 2 )